Delocalized Chern character for stringy orbifold K-theory

In this paper, we define a stringy product on $K^*_{orb}(\XX) \otimes \C $, the orbifold K-theory of any almost complex presentable orbifold $\XX$. We establish that under this stringy product, the de-locaized Chern character ch_{deloc} : K^*_{orb}(\XX) \otimes \C \longrightarrow H^*_{CR}(\XX), after a canonical modification, is a ring isomorphism. Here $ H^*_{CR}(\XX)$ is the Chen-Ruan cohomology of $\XX$. The proof relies on an intrinsic description of the obstruction bundles in the construction of Chen-Ruan product. As an application, we investigate this stringy product on the equivariant K-theory $K^*_G(G)$ of a finite group $G$ with the conjugation action. It turns out that the stringy product is different from the Pontryajin product (the latter is also called the fusion product in string theory).


INTRODUCTION
The notion of orbifold was first introduced by Satake under the name V-manifold. There have been many very interesting developments since its inception. For example, Kawasaki's orbifold index theory has been applied extensively in the study of geometric quantizations, and in the development of orbifold string theory in quantum physics.
Recall that an orbifold X is a paracompact Hausdorff space X equipped with a compatible system of orbifold atlases locally modeled on quotient spaces of Euclidean spaces by finite group actions. For each x ∈ X, there is a neighbourhood U x and a homeomorphism U x ∼ =Ũ x /G x . X is called effective if each local group G x acts onŨ x effectively. For any orbifold X, the de Rham cohomology H * dR (X) is well defined, and by a theorem of Satake [41], is isomorphic to the singular cohomology of the underlying space X = |X|.
Associated to an effective orbifold X, we have a canonical non-effective orbifold, called the inertia orbifoldX of X. The inertia orbifoldX consists of connected components of different dimensions, see page 7 in [14],X = (g) where (g) ∈ T 1 , the set of equivalence classes of conjugacy classes in local groups. Each X (g) is called a twisted sector of X, and is a sub-orbifold of X. The underlying topological space ofX, denoted by |X|, is the disjoint union of X and the singularity set where Conj(G x ) denotes the set of conjugacy classes in G x .
In the development of Gromov-Witten theory for symplectic orbifolds, Chen and Ruan in [14] discovered a remarkable new cohomology theory on any almost complex orbifold X, called the Chen-Ruan cohomology H * CR (X). Almost complex orbifolds are those orbifolds with local models given by a finite 2010 Mathematics Subject Classification. Primary: 57R19, 19L10, 22A22. Secondary: 55N15,53D45.
group acting unitarily on complex spaces. The Chen-Ruan cohomology H * CR (X), as a classical limit of an orbifold quantum cohomology, is a cohomology of the inertia orbifoldX H * CR (X) = (H * dR (X, C), • CR ) with a new product • CR , utilising the obstruction bundles over the moduli spaces of stable pseudoholomorphic orbifold curves in X. The obstruction bundle E [2] in the construction of the Chen-Ruan product is a complex orbifold vector bundle overX [2] =X × eX , where e :X → X is the immersion defined by the sub-orbifold structure on each connected component ofX. The associativity of the Chen-Ruan product follows from a property of the obstruction bundles discovered by Chen-Ruan in [14] using the gluing construction in Gromov-Witten theory. For a compact almost complex orbifold X, Adem, Ruan and Zhang in [4] defined a stringy product on K * orb (X, τ ), the twisted K-theory of the inertia orbifoldX with a transgressive twisting τ . This product will be called the Adem-Ruan-Zhang product, denoted by • ARZ . For an orbifold X arising from a smooth, projective variety M with an action of a finite group G or a Deligne-Mumford stack, an analogous product • JKK was defined by Jarvis, Kaufmann and Kimura in [24] on the untwisted orbifold K-theory ofX. The ring (K * orb (X), • JKK ) is called the full orbifold K-theory in [24]. This full orbifold K-theory was generalized to any compact abelian quotient orbifold X by Becerra and Uribe in [10]. The associated stringy product will be called the Becerra-Uribe produt, denoted by • BU . They also established an isomorphism In [10], Becerra and Uribe also established a ring homomorphism between the orbifold K-theory ofX and the Chen-Ruan cohomology under a modified Chern character map ch : (K * orb (X), • BU ) −→ (H * CR (X), • CR ).
We remark that this Chern character map is not an isomorphism over the complex coefficients. The full orbifold K-theory (K * orb (X), • JKK ) was further studied by Goldin, Harada, Holm and Kimura for abelian symplectic quotients in [20]. They gave a complete description of the ring structure of the full orbifold K-theory of weighted projective spaces obtained as symplectic quotients of C n by weighted S 1 -actions.
It is known that there is a delocalized Chern character (Cf. [8] for proper actions of discrete groups and [12] forétale groupoids) relating the orbifold K-theory K * orb (X) to the 2-periodic de Rham cohomology of the inertia orbifold X. In this paper, we define a stringy product on the orbifold K-theory K * orb (X) of an almost complex compact orbifold X. The main result of this paper is that after a canonical modification, ch deloc is an isomorphism over the complex coefficients, and sends the stringy product on K * orb (X) to the Chen-Ruan cup product on H * dR (X, C). The construction of this modified ch deloc relies on an intrinsic description of the obstruction bundle for the Chen-Ruan product and Adem-Ruan-Zhang's stringy product. Associated to the orbifold immersion e = (g) e (g) :X = (g) there is a 2-sector orbifoldX [2] =X × eX , which consists of a disjoint union of sub-orbifolds of X (g 1 ,g 2 ) X (g 1 ,g 2 ) each of which is labelled by an equivalence class of conjugacy pairs in local groups. Consider the following commutative diagram  I  I  I  I  I  I  I  I  I X (g 1 ) e (g 1 ) Then the obstruction bundle E [2] is a complex orbifold vector bundle overX [2] , whose component over each X (g 1 ,g 2 ) is denoted by E [2] (g 1 ,g 2 ) (see [14] for its definition). Over the inertia orbifoldX, there is an orbifold complex vector bundle given by the orbifold normal bundle of each X (g) in X, each of which admits an automorphism of finite order. We can choose a Hermitian metric on the tangent bundle of X such that the automorphism Φ acts unitarily on N e . Then each N (g) has an eigen-bundle decomposition as linear combinations of vector bundles with rational coefficients, or as elements in Theorem 1.1. Given an almost complex orbifold X, let N be the normal bundle ofX [2] in X. Then the obstruction bundle E [2] in [14] satisfies the following identity Theorem 1.1 implies that the linear combination of vector bundle with rational coefficients after combining like terms, is a genuine vector bundle, which can be identified with the obstruction bundle E [2] . This theorem was obtained in [13] for abelian orbifolds and in [24] for smooth Deligne-Mumford stacks. In this paper, we give a direct proof of this theorem using an equivariant version of Kawasaki's orbifold index theorem. We also employ this theorem to give an intrinsic definition of Chen-Ruan cohomology as Chen and Hu did for abelian orbifolds [13].
The modified delocalized Chern character is given by where on each component X (g) , T(N e , Φ) is defined by associated to the eigen-bundle decomposition (1.1). Here T(V ) m is the multiplicative characteristic class of an orbifold complex vector bundle V corresponding to the formal power series ( is a ring isomorphism. In Section 2, we review some basics of orbifold, orbifold K-theory and orbifold index theorem used in this paper. In Section 3, we give an intrinsic definition of the Chen-Ruan cohomology after we establish Theorem 1.1. In Section 4, we define the stringy product on orbifold K-theory of an almost complex compact orbifold, and prove Theorem 1.2. We also compute a few examples in Section 4, and discover that the stringy product on the orbifold K-theory of the orbifold [G/G] (obtained from the conjugation of a finite group on itself) is different from the Pontrjagin product on K G (G). In Section 5, we briefly discuss the stringy product on twisted orbifold K-theory for orbifolds with torsion twistings.

REVIEW OF ORBIFOLDS AND ORBIFOLD INDEX THEORY
In this section, we will give a brief review of the notion of orbifolds in terms of orbifold atlas and properétale groupoids. Then we review the orbifold index theory using delocalized Chern character. The main references for this section are [2], [25], [28], [30] and [42].

Orbifolds and orbifold groupoids.
Let X be a paracompact Hausdorff space. An n-dimensional orbifold chart for an open subset U of X is a triple (Ũ , G, π) given by a connected open subsetŨ ⊂ R n , together with a smooth action of a finite group G such that π :Ũ → U is the induced quotient map. An orbifold chart (Ũ , G, π) is called effective if the action of G onŨ is effective. Given an inclusion ι ij : U i ֒→ U j , an embedding of orbifold charts consists of an injective group homomorphism λ ij : G i → G j , and an embedding φ ij :Ũ i ֒→Ũ j covering the inclusion ι ij such that φ ij is G i -equivariant with respect to φ ij , that is, for x ∈Ũ i and g ∈ G i . In the noneffective case, we further require that the subgroup of G i acting trivially on U i is isomomorphically mapped to the subgroup of G j acting trivially on U j . Whenever U i ⊂ U j ⊂ U k , there exists an element g ∈ G k such that (2) Given any inclusion U i ⊂ U j , there is an embedding of orbifold charts (φ ij , λ ij ) : Two orbifold atlases U and V are equivalent if there is a common orbifold atlas W refining U and V.
An orbifold X = (X, U ) is a paracompact Hausdorff space X with an equivalence class of orbifold atlases. Given an orbifold X = (X, U ) and a point x ∈ X, let (Ũ , G, π) be an orbifold chart around x. Then the local group at x is defined to be the stabilizer ofx ∈ π −1 (x), which is uniquely defined up to conjugation.
The notion of an orbifold and many of its invariants can be reformulated using the language of groupoids. For general orbifolds, the groupoid viewpoint is also essential for the C * -algebraic definition of K-theory and its twisted version in order to get a cohomology theory satisfying the Mayer-Vietoris axiom. We briefly recall the definition of groupoids and their roles in the orbifold theory.
A Lie groupoid G = (G 1 ⇉ G 0 ) consists of two smooth manifolds G 1 and G 0 , together with five smooth maps (s, t, m, u, i) satisfying the following properties.
(1) The source map and the target map s, t : written as m(g 1 , g 2 ) = g 1 · g 2 , satisfies the obvious associative property.
(3) The unit map u : G 0 → G 1 is a two-sided unit for the composition. (4) The inverse map i : G 1 → G 1 , i(g) = g −1 , is a two-sided inverse for the composition.
A Lie groupoid G = (G 1 ⇉ G 0 ) is proper if (s, t) : G 1 → G 0 × G 0 is proper, and calledétale if s and t are local diffeomorphisms.
Let G 1 ⇉ G 0 and H 1 ⇉ H 0 be two Lie groupoids. A generalized morphism between G and H is a right principal H-bundle P f over G 0 which is also a left G-bundle over H 0 such that the left G-action and the right H-action commute, formally denoted by For example, a rank k Hermitian vector bundle over a Lie groupoid G is defined by a U (k)-valued cocycle over G, that is, a generalized morphism G → U (k) where U (k) is viewed as a Lie groupoid with one object. Note that generalized morphisms can be composed. This implies that the pull-back of a vector bundle over a groupoid by any generalized morphism is well-defined. Note that a generalized morphism f between G and H is invertible if P f in (2.1) is also a principal G-bundle over H 0 . Then G and H are called Morita equivalent.
(1) As observed in [31] and [27], given an orbifold X = (X, U ), there is a canonical properétale Lie groupoid G[U ], locally given by the action groupoidŨ i ⋊ G i ⇉Ũ i . For two equivalent orbifold atlases U and V, G[U ] and G[V] are Morita equivalent.
(2) Given a properétale Lie groupoid G, there is a canonical orbifold structure on its orbit space |G|, see [30] and Proposition 1.44 in [2]. Two Morita equivalent properétale Lie groupoids define the same orbifold up to isomorphism (Theorem 1.45 in [2]). (3) Given an orbifold X = (X, U ), a properétale Lie groupoid G is called a presentation of X if there is a homeomorphism f : |G| → X such that f * U agrees with the canonical orbifold structure on |G|. A properétale Lie groupoid is also called an orbifold groupoid for simplicity.
(4) An orbifold is called effective if any local group acts effectively on its orbifold chart. For an n-dimensional effective orbifold X = (X, U ), the corresponding properétale Lie groupoid is Morita equivalent to the action groupoid associated to the O(n)-action on the orthonormal frame bundle for a Riemannian metric on X.
Definition 2.3. Let X be an orbifold with a presenting groupoid G. If |G| is compact, the de Rham cohomology of an orbifold X = (X, U ), denoted by H * orb (X, R), is defined to be the de Rham cohomology of G which is the cohomology of the G-invariant de Rham complex (Ω p (G), d), where If |G| is not compact, the de Rham cohomology of X is defined to be the de Rham cohomology with compact supports of G, where a differential form ω ∈ Ω p (G) has compact support in |G|.
An orbifold vector bundle E over an orbifold X = (X, U ) is a family of equivariant vector bundles such that for any embedding of orbifold charts φ ij : (Ũ i , G i ) ֒→ (Ũ j , G j ), there is a G i -equivariant bundle mapφ ij :Ẽ i →Ẽ j covering φ ij :Ũ i →Ũ j . The total space E = (Ẽ i /G i ) of an orbibundle E → X has a canonical orbifold structure given by {(Ẽ i , G i )}. A connection ∇ on an orbibundle E → X is a family of invariant connections {∇ i } on {(Ẽ i →Ũ i , G i )} which are compatible with the bundle maps {φ ij }. Examples of orbibundles over X include its tangent bundle T X and its cotangent bundle T * X. In general, orbifold vector bundles over X do not define vector bundles over the underlying topological space X.
Recall that a vector bundle over a Lie groupoid G = (G 1 ⇉ G 0 ) is a G-vector bundle E over G 0 , that is, a vector bundle E with a fiberwise linear action of G covering the canonical action of G on G 0 . One can check that an orbibundle E = (E, U E ) defines a canonical vector bundle over the groupoid G[U ]. Definition 2.4. Let X be an orbifold with a presenting groupoid G. If |G| is compact, the orbifold Ktheory of X, denoted by K 0 orb (X), is defined to be the Grothendieck ring of isomorphism classes of vector bundles over G. If |G| is not compact, the orbifold K-theory of X is defined to be the Grothendieck ring of isomorphism classes of complex vector bundles with compact supports over G. Here a vector bundle with compact support over G is a Z 2 -graded G-equivariant complex vector bundle E = E 0 ⊕ E 1 with a G-equivariant bundle morphism σ : E 0 → E 1 such that the support of σ {x ∈ G 0 |σ x : E 0 x → E 1 x is not an isomorphism} defines a compact set in |G|.
The de Rham cohomology and the orbifold K-theory are well-defined, as a Morita equivalence between two groupoids induces an isomorphism on de Rham cohomology and orbifold K-theory respectively. The Satake-de Rham theorem for an orbifold X = (X, U ) leads to an isomorphism between the de Rham cohomology and the singular cohomology of the underlying topological space. The standard Chern-Weil construction applied to a compact orbifold X gives rise to the Chern character map ch : K 0 orb (X) −→ H ev orb (X, C), which is a ring homomorphism. Here the ring structure on K 0 orb (X) is given by the tensor product of vector bundles and the ring structure on H ev orb (X, C) is given by the wedge product of differential forms. This Chern character over the complex coefficients is not an isomorphism.
Remark 2.5. The definition of orbifold K-theory can be extended in the usual way to a Z 2 -graded cohomology theory, see [2], For an orbifold as a quotient of a compact Lie group action on locally compact manifolds with finite stabilizers, the orbifold K-theory has the usual Bott periodicity, the Mayer-Vietoris exact sequence and the Thom isomorphism for orbifold Spin c vector bundles.
We can define the orbifold K-theory of an orbifold X as the K-theory of the reduced C * -algebra C * red (X) of the canonical properétale groupoid (see Chapter 2 in [15]). The C * -algebraic orbifold Ktheory is a module over the orbifold K-theory using orbifold vector bundles. If X is presentable, that is, X is the orbifold obtained from a locally free action of a compact Lie group G on a smooth manifold M , then C * (X) is Morita equivalent to the cross-product C * -algebra C(M ) ⋊ G. This Morita equivalence can be used to define an isomorphism between the orbifold K-theory K * orb (X) defined in Definition 2.3 and the K-theory of the reduced C * -algebra C * red (X) Conjecturally, the isomorphism K * orb (X) ∼ = K * (C * red (X)) holds for general orbifolds.

Delocalized Chern character and the orbifold index theory.
For any compact orbifold X, there is a delocalized Chern character, from the orbifold K-theory of X to the 2-periodic de Rham cohomology of its inertia orbifoldX. It was defined in [8] for proper actions of discrete groups and in [12] forétale groupoids. The delocalized Chern character made its first appearance in the Lefschetz formulas of Atiyah-Bott in [5] and the Kawasaki orbifold index theorem in [25]. Let X = (X, U ) be an orbifold. Then the set of pairs where (g) Gx is the conjugacy class of g in the local group G x , has a natural orbifold structure given by Here for each orbifold chart (Ũ , G, π, U ) ∈ U , Z G (g) is the centralizer of g in G andŨ g is the fixedpoint set of g inŨ . This orbifold, denoted byX, is called the inertia orbifold of X. The inertia orbifold X consists of a disjoint union of sub-orbifolds of X.
To describe the connected components ofX, we need to introduce an equivalence relation on the set of conjugacy classes in local groups as in [14]. For each x ∈ X, let (Ũ x , G x , π x , U x ) be a local orbifold chart at x. If y ∈ U x , then up to conjugation, there is an injective homomorphism of local groups G y → G x , hence the conjugacy class (g) Gx is well-defined for g ∈ G y . We define the equivalence to be generated by the relation (g) Gy ∼ (g) Gx . Let T 1 be the set of equivalence classes. Theñ Note that X (1) = X is called the non-twisted sector and X (g) for g = 1 is called a twisted sector of X.
Let G be a properétale Lie groupoid representing a compact orbifold X = (X, U ). Then the groupoid associated to the inertia orbifoldX is given byG = ((s, t) : with the source map s(g, h) = g and the target map t(g, h) = h −1 gh. There is an obvious evaluation map e :G → G which corresponds to the obvious orbifold immersion Given a complex orbifold bundle E over X, or a complex vector bundle over its presenting groupoid G, the pull-back bundle e * E overX orG has a canonical automorphism Φ. With respect to a G-invariant Hermitian metric on E, there is an eigen-bundle decomposition of e * E e * E = where E θ is a complex vector bundle overG, on which Φ acts by multiplying e 2π √ −1θ . Define The odd delocalized Chern character can be defined in the usual way. Using the standard compactly supported condition, the delocalized Chern character can be defined for non-compact orbifolds.
where the closed submanifold M g is the fixed point set of the g-action, and Z G (g) is the centralizer of g in G. Applying the ordinary Chern character on each K * (M g /Z G (g)), we get an alternative definition of the delocalized Chern character over C: Going through the proof of Theorem 5.1 in [3], particularly the ring map from the representation ring of the cyclic subgroup g generated by g to the the cyclotomic field Q(e 2π √ −1/|(g)| ), one can check that the delocalized Chern character (2.2) agrees with the delocalized Chern character defined by eigen-bundle decompositions. Here we identify the inertia orbifoldX with the orbifold g [M g /Z G (g)].
Proposition 2.7. For any compact presentable orbifold X, the delocalized Chern character gives a ring isomorphism Proof. The proof follows from the isomorphism for orbifolds obtained from a finite group action on a locally compact manifold and the Mayer-Vietoris sequence for open covers. Recall that the canonical groupoid G associated to an orbifold chart {(Ũ i , G i , π i )}, when restricted to each open setŨ i , is an action groupoidŨ i ⋊ G i . With compactly supported K-theory and de Rham cohomology, one has the following isomorphism (Theorem 1.19 in [8]) of vector spaces over C From the definition, we see that H * c (Ũ i , G i ) is the de Rham cohomology of the inertia groupoid of the action groupoidŨ i ⋊ G i . By an induction argument, we can apply the Mayer-Vietoris sequence for open covers and the five lemma to show that is an isomorphism of vector spaces. For two vector bundles E and F with eigen-bundle decompositions we have from which one immediately gets Hence, ch deloc : is a ring isomorphism. The ring isomorphism for the odd delocalized Chern character can be proved by the standard de-suspension argument.
Remark 2.8. The delocalized Chern character can be applied to write the Kawasaki orbifold index [25] as follows. Let X be a compact almost complex orbifold with a Hermitian connection on the tangent bundle TX of the inertia orbifoldX. Let E be a complex orbifold Hermitian vector bundle with a Hermitian connection. Let / D ± E be the corresponding Spin c Dirac operator. The orbifold index formula in [25] can be expressed as where T d deloc (X) ∈ H * (X) is the delocalized Todd class of X, whose X (g) -component is given by Here T d(X (g) ) is the Todd form of X (g) and F N (g) is the curvature of the induced Hermitian connection on the normal bundle N (g) . In this paper, we will use the equivariant orbifold index formula for a finite group H acting trivially on X. Here we briefly discuss this version of the equivariant orbifold index formula. When H acts trivially on X, we have where R(H) is the representation ring of H. Composing this isomorphism with the delocalized Chern character, we get

INTRINSIC DESCRIPTION OF CHEN-RUAN COHOMOLOGY
For an almost complex abelian orbifold X, Chen and Hu gave a classical definition of the Chen-Ruan product using an intrinsic definition of Chen-Ruan's obstruction bundle, see the proof of Proposition 1 in section 3.4 of [13]. For a smooth Deligne-Mumford stack, a similar description of Chen-Ruan's obstruction bundle was obtained by Jarvis, Kaufmann and Kimura in [24]. In this section, we give an intrinsic definition of Chen-Ruan's obstruction bundle for an almost complex orbifold X with associated Lie groupoid G.
LetX = (g)∈T 1 X (g) be the inertia orbifold with associated inertia groupoidG and the evaluation map e :G → G. The k-sectorX [k] of X is defined to be the orbifold on the set of all pairs where (g 1 , · · · , g k ) Gx denotes the conjugacy class of k-tuples. Here two k-tuples (g consists of a disjoint union of sub-orbifolds of X where T k denotes the set of equivalence classes of conjugacy k-tuples in local groups. Then the groupoid G [k] associated to the k-sectorX [k] for k ≥ 2 is given bỹ with the source map s(g 1 , g 2 , · · · , g k , h) = (g 1 , g 2 , · · · , g k ) and the target map t(g 1 , g 2 , · · · , g k , h) = (h −1 g 1 h, h −1 g 2 h, · · · , h −1 g k h).
As in [14],G [k] can be identified with the orbifold moduli space of constant pseudo-holomorphic maps from an orbifold sphere with k + 1 orbifold points to X. The obstruction bundle over the orbifold moduli space, given by the cokernel of the Cauchy-Riemann operator over the orbifold sphere coupled with the pull-back of the tangent bundle T G of G, defines a complex vector bundle E [k] over the groupoid of k-sectorsG [k] , for k ≥ 2.
Definition 3.1. Let E be any complex vector bundle with an automorphism Φ of finite order over a properétale groupoid G. Choose a Hermitian metric on E preserved by Φ. Then E has an eigen-bundle decomposition as a linear combination of vector bundles with rational coefficients, or an element in K 0 (G) ⊗ Q.
Given an almost complex orbifold X, any orbifold complex vector bundle over a compact orbifoldX, or equivalently any vector bundle overG, has an automorphism. Specifically, if E is a vector bundle over G, then the fiber E g at g ∈G 0 has a linear isomorphism induced by the action of g. In particular, e * T G is a complex vector bundle overG with an automorphism Φ, and TG is a sub-bundle of e * T G on which Φ acts trivially. Let N e be the normal bundle of the evaluation map e with the induced automorphism Φ. We can choose a G-invariant Hermitian metric on G such that Φ acts unitarily on N e . So the automorphism preserves the orthogonal decomposition As a complex vector bundle overG, N e admits an eigen-bundle decomposition Consider the following commutative diagram Let N be the normal bundle to the map e • e 12 = e • e 1 = e • e 2 :G [2] → G. Then N as a complex vector bundle overG [2] , has three automorphisms here N e 1 is the normal bundle to the map e 1 . Then N Φ 1 , by Definition 3.1, is given by (2) N ∼ = N e 2 ⊕ e * 2 N e with an automorphism Φ 2 = Id ⊕ e * 2 Φ, here N e 2 is the normal bundle to the map e 2 . Then we have (3) N ∼ = N e 12 ⊕ e * 12 N e with an automorphism Φ 12 = Id ⊕ e * 12 Φ, here N e 12 is the normal bundle to the map e 12 . Then we have When X is compact, all of these three automorphisms have finite order.
The main result of this section is the following intrinsic description of the obstruction bundle E [2] → G [2] in the definition of the Chen-Ruan cohomology. This generalises the results of [13] and [24] to general almost complex orbifolds. Our proof using the orbifold index theorem was inspired by the proof in [13] for abelian cases, and is simpler than the proof given in [24] for smooth Deligne-Mumford stacks.
Theorem 3.2. The obstruction bundle E [2] in the construction of the Chen-Ruan product satisfies the following identity Proof. We first recall the definition of the obstruction bundle from Chapter 4.3 in [2] and Section 4 in [14]. IdentifyG [2] as the moduli space of constant representable orbifold morphisms from the orbifold Riemann sphere S 2 with three orbifold points to X. Given a point (g 1 , g 2 ) ∈G [2] 0 with s(g i ) = t(g i ) = x, let N be a finite subgroup generated by g 1 , g 2 in the local group By Lemma 4.5 in [2], up to an isomorphism, N depends only on the connected component ofX [2] , the orbifold associated toG [2] . Let m 1 , m 2 and m 3 be the order of g 1 , g 2 and g 1 g 2 respectively. There is a smooth compact Riemann surface Σ such that Σ/N is an orbifold Riemann sphere (S 2 , (m 1 , m 2 , m 3 )) with three orbifold points of multiplicities (m 1 , m 2 , m 3 ). Note that Σ is given by the quotient of the orbifold universal cover of (S 2 , (m 1 , m 2 , m 3 )) by the kernel of the surjective homomorphism (Cf. (4.14) in [2]) The constant orbifold morphism corresponding to (x, g 1 , g 2 ) is represented by an ordinary constant mapf x : Σ →Ũ x for an orbifold chart (Ũ x , G x ) at x. The elliptic complex for the obstruction bundle over the point (x, g 1 , g 2 ) is the N -invariant part of the elliptic complex So the tangent space of the moduli spaceG [2] at (g 1 , g 2 ) ∈G [2] 0 is given by and the obstruction bundle over the point (g 1 , g 2 ) ∈G [2] 0 is given by Here Z Gx (g 1 ) and Z Gx (g 2 ) are the centralizers of g 1 and g 2 respectively in the local group G x . Note that (e • e 12 ) * T x G = N (g 1 ,g 2 ) ⊕ T (g 1 ,g 2 )G [2] and N acts on T (g 1 ,g 2 )G [2] trivially. We have Combining with (3.2), this leads to Applying the orbifold index formula (Proposition 4.2.2 of [14]) to the N -invariant part of the elliptic complex∂ Σ | T (g 1 ,g 2 )G [2] : Ω 0 (Σ, we get Here the contributions to the orbifold index formula from the three singular points are all zero due to the trivial action of N on T (g 1 ,g 2 )G [2] . As representations, which can be glued together using the groupoid action to form a complex vector bundle E [2] over the groupoidG [2] . Let Note that the automorphisms Φ 1 , Φ 2 and Φ −1 12 , induced by the action of g 1 , g 2 and (g 1 g 2 ) −1 respectively, preserve each irreducible component V λ in (3.5), and commute with the action of N . Any eigen-subspace of V λ of Φ 1 , Φ 2 or Φ −1 12 is also a representation of Z Gx (g 1 ) ∩ Z Gx (g 2 ). The irreducibility of V λ implies that 12 acts on V λ as multiplication by e 2π √ −1θ λ 12 for θ 12 ∈ (0, 1) ∩ Q.
Note that Z Gx (g 1 ) ∩ Z Gx (g 2 ) acts trivially the orbifold sphere (S 2 , (m 1 , m 2 , m 3 )). Applying the equivariant version of Kawasaki's orbifold index formula (See Remark 2.8) to the N -invariant part of the elliptic complex∂ . This implies that, as a representation of Z Gx (g 1 ) ∩ Z Gx (g 2 ), after combining like terms. Taking the direct sum over λ as in (3.5), we obtain in the representation ring of Z Gx (g 1 ) ∩ Z Gx (g 2 ) over the rational coefficients Q.
The normal bundle N is a complex vector bundle over the groupoidG [2] so the fiberwise identity (3.8) leads to the identity E [2] ⊕ N = e * 1 N e,Φ + e * 2 N e,Φ + e * 12 N e,Φ −1 in K 0 (G [2] ) ⊗ Q. This completes the proof of the theorem. (1) One particular consequence of Theorem 3.2 is that the linear combination of vector bundles with rational coefficients e * 1 N e,Φ + e * 2 N e,Φ + e * 12 N e,Φ −1 − N , after combining like terms, is a complex vector bundle which can be identified with the obstruction bundle E [2] in the construction of the Chen-Ruan product. (2) In [22], Hepworth also provides another description of the obstruction bundle. As remarked in the introduction of [22], his description of the obstruction bundle involves the theory of orbifold Riemann surfaces to get certain inequalities regarding the degree shifting in the definition of the Chen-Ruan cohomology.
With this intrinsic description of the obstruction bundle, we can extend Chen-Hu's alternative definition of the Chen-Ruan cohomology ring for abelian orbifolds to general orbifolds by identifying with certain formal cohomology classes on X. We firstly recall the definition of the Chen-Ruan cohomology of an almost complex, compact orbifold. More details can be found in [14].

Definition 3.4. The Chen-Ruan cohomology (H
with a degree shift and a new product * CR given by the following formula, for ω 1 , ω 2 ∈ H * (X, C), where e(E [2] ) ∈ H * (X [2] , C) is the cohomological Euler class of the obstruction bundle E [2] . The degree shifting number ι : (g)∈T 1 X (g) → Q is defined to be a locally constant function onX.
Given an oriented real orbifold vector bundle V over X, or equivalently, an oriented vector bundle V over G, there is a compactly supported differential form for any differential form α on V , and i : X → V is the inclusion map defined by the zero section. Such a differential form defines the Thom class Θ(V ) of V .
Note that the degree of Θ(N (g) ) is 2n (g) , where n (g) is the complex codimension of X (g) in X. The homomorphism H * +2n (g) (N (g) ) (3.10) sending ω (g) ∈ H * (X (g) ) to π * (g) ω (g) ∧ Θ(N (g) ), is the Thom isomorphism. Here π (g) : N e (g) → X (g) is the bundle projection. Identifying a neighbourhood of the zero section in N e ( g) containing the support of Θ(N e ( g) ), with the orbifold neighbourhood ofX (g) in X, we obtain the push-forward map As before, let be the eigen-bundle decomposition for the automorphism Φ.
Definition 3.6. The fractional Thom class of the normal bundle (N (g) , Φ) over X (g) is defined by the formal wedge product of formal degree 2ι (g) = θ (g) θ (g) , with compact support in a neighborhood of X (g) in X.
For each (g) ∈ T 1 , denote by H * +2ι (g) Φ (X) the set of formal wedge products Elements in H * +2ι (g) Φ (X), called formal cohomology classes, can be represented by formal products of closed differential forms on X and a formal fraction of differential forms for the fractional Thom class. These formal differential forms and their formal cohomology classes have the wedge product obtained from the formal product. We use the convention that integration of a formal differential form over X vanishes unless its formal degree agrees with the dimension of X. The fractional Thom class Θ(N e , Φ) can be employed to define a formal push-forward map obtained from the Thom isomorphism (3.10) with the usual Thom class replaced by the corresponding fractional Thom class. It is clear from the definition that e Φ * is injective. Similarly, we can define another formal push-forward map with the fractional Thom class Θ(N (g) , Φ) replaced by The following lemma explains the role of these two formal push-forward maps.
Lemma 3.7. The Thom class of the normal bundle N (g) → X (g) is given by Θ (N (g) , Φ) ∧ Θ(N (g) , Φ −1 ). Moreover, For α ∈ H * (X (g) ) and β ∈ H * (X (g −1 ) ), we have Proof. Note that N (g) = θ (g) N (θ (g) ), from which we get . From the definition of orbifold integration, we have Here π (g) : N e (g) → X (g) is the bundle projection and N e (g) is identified with a neighbourhood of X (g) in X.
By Poincáre duality for orbifolds and the 3-point function (3.9) for the Chen-Ruan product, we get

STRINGY PRODUCT ON THE ORBIFOLD K-THEORY
In this section, we will define a stringy product on the orbifold K-theory of an almost compact orbifold X, and establish a ring isomorphism between the orbifold K-theory of X and the Chen-Ruan cohomology ring using a modified version of the delocalized Chern character. We first recall the definition of the delocalized Chern character and its properties. Then we give a geometric definition of a stringy product on K * orb (X) ⊗ C itself and show that this product agrees with the Chen-Ruan product under a modified version of the delocalized Chern character.

4.1.
Review of the Adem-Ruan-Zhang stringy product. In [4], Adem, Ruan and Zhang defined a stringy product on the twisted K-theory of the inertia orbifoldX of a compact, almost complex orbifold X. This product will be called the Adem-Ruan-Zhang product, denoted by • ARZ . To simplify the construction, we first discuss the untwisted case for a presentable, compact, almost complex orbifold X.
The Adem-Ruan-Zhang product on the orbifold K-theory of the inertia orbifoldX, under the canonical isomorphism for a presentable orbifold X is given by the formula [2] ] ∈ K * (G [2] ) is the K-theoretical Euler class of the obstruction bundle E [2] . We recall here that the push-forward map is given by the composition of the Thom isomorphism and the natural extension homomorphism K * c (N e 12 ) → K * (G) obtained by identifying the normal bundle N e 12 as a (component-wise) tubular neighbourhood ofG [2] inG. Under the decomposition for α 1 ∈ K * orb (X (g 1 ) ) and α 2 ∈ K * orb (X (g 2 ) ) we have For an abelian almost complex orbifold X obtained from a compact action of an abelian Lie group on a compact manifold, Becerra and Uribe in [10] established a ring homomorphism by modifying the usual Chern character as in [24]. The decomposition theorem of Adem-Ruan (Theorem 5.1 in [3]) implies that this ring homomorphism is not an isomorphism in general. Now we briefly recall the definition of twisted K-theory for orbifolds. A twisting on an orbifold groupoid G is a generalized morphism where P U (H), viewed as a groupoid with the unit space {e}, is the projective unitary group P U (H) of an infinite dimensional separable complex Hilbert space H, with the norm topology. A twisting σ gives rise to a principal P U (H)-bundle P σ over G. Two twistings are called equivalent if their associated P U (H)-bundles are isomorphic.
Remark 4.1. It was argued in [6] that it is not desirable to use the norm topology on the structure group P U (H), particularly in dealing with equivariant twisted K-theory. For orbifolds and orbifold groupoids, it suffices to consider P U (H) with the norm topology in the definition of twistings and twisted K-theory just as in the non-equivariant case. The main reason is that the underlying groupoid for any orbifold is proper andétale, so locally it is given by the transformation groupoid for a finite group action. As pointed out in [6], the structure group P U (H) with the norm topology works fine for an almost free action of a compact Lie group, where the underlying orbifold is presentable.
Let K(H) be space of compact operators on H endowed with the norm topology, and let F red(H) be the space of Fredholm operators endowed with the * -strong topology and F red 1 (H) be the space of self-adjoint elements in F red(H). Consider the associated bundles over the groupoid G. The σ-twisted K-theory of G, denoted K i (G, σ) for i = 0, 1, is defined to be the set of homotopy classes of compactly supported sections of F red i (P σ ). For any orbifold X = (X, U ), the twisted K-theory of X is defined to be the twisted K-theory of the associated properétale groupoid G[U ]. Then the twisted orbifold K-theory is a module over the ordinary orbifold K-theory and satisfies the Mayer-Vieroris sequence and the Thom isomorphism for complex orbifold vector bundles. Moreover, the twisted orbifold K-theory admits the multiplicative operation where the addition σ 1 + σ 2 is the new twisting from the group homomorphism P U (H) × P U (H) → P U (H) induced by the Hilbert space tensor product H ∼ = H ⊗ H. See [44] [27] [43] for more detailed discussions.

Stringy product on orbifold K-theory.
Let X be an almost complex compact orbifold, andX = (g) X (g) be its inertia orbifold. Let G andG be their presenting properétale groupoids. The evaluation map e = ⊔ (g) e (g) : (g) X (g) → X is presented by the map e :G → G.

Proposition 4.2.
There exists a canonical ring homomorphism such that the following diagram Proof. Any complex orbifold vector bundle E over the inertia orbifoldX admits an automorphism Φ. In terms of the associated complex vector bundle E over the canonical properétale groupoidG, the action of Φ on each fiber E g over the point g ∈G 0 is given by the action of g. We have an eigen-bundle where E θ is a complex vector bundle overG, on which Φ acts as multiplication by e 2π √ −1θ . Then defines a homomorphism ch Φ : K 0 orb (X) −→ H ev (X, C). From the definition of the delocalized Chern character, we get ch deloc = ch Φ • e * . Hence, the diagram (4.5) commutes. Proposition 4.2 implies the following commutative diagram where ch deloc is an isomorphism. For any elementα ∈ Im(e * ) ∩ Ker(ch Φ ), there exists α ∈ K * orb (X) ⊗ C such thatα = e * (α) and ch Φ (α) = 0. These imply that ch deloc (α) = 0. Hence α = 0 as ch deloc is an isomorphism. Theñ α = e * (α) = e * (0) = 0. Then we obtain Hence, each elementα ∈ K * orb (X) ⊗ Z C can be uniquely written as for a unique element α ∈ K * orb (X) ⊗ Z C and β ∈ Ker(ch Φ ). Define e # : K * orb (X) ⊗ Z C −→ K * orb (X) ⊗ Z C, (4.9) sendingα = e * α + β as in (4.8) to α. Then one can check that e # is a left inverse of e * such that ch deloc • e # (α) = ch Φ (α) (4.10) for anyα ∈ K * orb (X) ⊗ Z C. With these preparations, we now can define the stringy product on the orbifold K-theory of a compact almost complex orbifold X. For simplicity, we will denote K * orb (X) ⊗ Z C and K * orb (X) ⊗ Z C by K * orb (X, C) and K * orb (X, C) respectively.

Definition 4.3.
Let X be an almost complex compact orbifold, andX = (g) X (g) be its inertia orbifold. The stringy product • on K * orb (X, C) is defined by for α 1 , α 2 ∈ K * orb (X, C). Here e * α 1 • ARZ e * α 2 is the Adem-Ruan-Zhang stringy product on K * orb (X, C) and e # is the left inverse of e * .
Next, we define a modified version of the delocalized Chern character For a complex vector bundle E over a compact manifold M , there is a well-known Chern character defect for Thom isomorphisms in K-theory and cohomology theory, for example see Chapter III.12 in [26], such that the following diagram commutes. Here T(E) is a characteristic class of E associated to the formal power series That is, if we formally split the total Chern class as c(E) = (1 + x j ), then Given an orbifold complex vector bundle over the inertia orbifoldX, let be the eigen-bundle decomposition of the canonical automorphism Φ, where Φ acts on E m i as multiplication by e 2π √ −1m i . Define a cohomology class where T(E m i ) m i is the characteristic class associated to the formal power series 1 − e x x m . Then T(E, Φ) is an invertible element in H * (X, C) as the degree zero component is 1.
Associated to the normal bundle , we have the cohomology class T(N e , Φ) in H * (X) whose component in H * (X (g) , C) is given by T (N (g) , Φ). The intrinsic description of the obstruction bundles overX [2] in Theorem 3.2 gives rise to the following identity in H * (X (g 1 ,g 2 ) , C) for any connected component X (g 1 ,g 2 ) ofX [2] . Here N (g 1 ,g 2 ) is the normal bundle for the orbifold embedding e 12 : X (g 1 ,g 2 ) → X (g 1 g 2 ) .
Definition 4.4. The modified delocalized Chern character on the orbifold K-theory K * orb (X) ∼ = K * (G) is defined to be Now we can prove the main result of this paper. is a ring isomorphism between two Z 2 -graded multiplicative cohomology theories.
Here we apply the identity (4.12). This implies that ch deloc is a ring homomorphism.
From Proposition 2.7, we know that ch deloc : K * orb (X) ⊗ C → H * CR (X, C) is an isomorphism of complex vector spaces. As the degree zero component of T(N e , Φ) is 1, T(N e , Φ) is an invertible element of the ring H * CR (X, C). So ch deloc is also an isomorphism of complex vector spaces. Hence ch deloc is a ring isomorphism.
Remark 4.6. Note that K * orb (X) = (g) K * orb (X (g) ), H * (X, C) = (g) H * (X (g) , C), and ch Φ : preserves the decompositions. This decomposition preserving homomorphism motivates the following alternative definition of stringy product which might simplify computation when applying Theorem 4.5. The isomorphism ch deloc : K * orb (X, C) −→ (g) H * orb (X (g) , C) induces the decomposition such that ch deloc preserves the decomposition. Then the commutative traingle (4.6) immediately implies the following identities for any ω (g) ∈ K * orb X, (g) , and where e (h) : X (h) → X denotes the orbifold embedding. Definē by sending α (g) ∈ K * orb X, (g) to e * (g) α (g) ∈ K * orb (X (g) , C). It is straightforward to check thatē * is a ring homomorphism preserving the decomposition. Moreover, the homomorphismē * satisfies ch deloc = ch Φ •ē * and has a canonical left inverseē # defined in a similar way as in (4.8) -(4.10). We can check that the stringy product in Definition 4.3 can be written as for α 1 , α 2 ∈ K * orb (X, C). In particular, if α 1 ∈ K * orb X, (g 1 ) and α 2 ∈ K * orb X, (g 2 ) , then for χ 1 , χ 2 ∈ C(G), and g ∈ G. This means The stringy product on the orbifold K-theory of the inertia orbifold was described in [4]. Here we consider the stringy product on K G ({pt}), the orbifold K-theory of the orbifold [{pt}/G] itself. Note that that is, ch deloc : R(G) → C is a C-valued class function on G. Then a ring isomorphism, where the ring structure on R(G) ⊗ C is given by the tensor product of representations, and the ring structure on C(G) is the standard point-wise multiplication. Let ρ V : G → GL(V ) and ρ W : G → GL(W ) be two representations of G. We compute the Adem-Ruan-Zhang product represented by a G-equivariant vector bundle over G, since there are no normal bundles involved in this case. As a vector bundle overG, its fiber over g ∈ G is where V g 1 = V and W g 2 = W . The automorphism Φ on the fiber over g is given by By a direct calculation, we get where * is the convolution product on C(G). Therefore, the stringy product on K orb ([{pt}/G], C) agrees with the convolution product on C(G) under the delocalized Chern character. Therefore, we get the ring isomorphism  where Z G (g) is the centralizer of g in G, see [4]. An element in K G (G) is represented by a collection of finite dimensional complex vector spaces {(V g , ρ g )} g∈G , where ρ g : Z G (g) → GL(V g ) is a group homomorphism, such that there is a linear G-action on ⊕ g V g intertwining with the ⊕ρ g -representation in the following sense: for any h ∈ Z G (g).
The diagram (4.19) implies that for (g, h) ∈G T r ρ g (h) = T r ρ kgk −1 (khk −1 . Hence the function χ {(Vg ,ρg)} g∈G : (g, h) → T r ρ g (h) is a G-invariant function onG. Let C G (G) be the space of C-valued G-invariant functions onG. Then the delocalized Chern character There are three different products on K G (G) ⊗ Z C described as follows.
(1) The first one is given by the usual tensor product ⊗ G of G-equivariant vector bundles over G.
The delocalized Chern character is a ring isomorphism for the point-wise product · on C G (G). (2) The second one is the Pontryagin product • G given by where π 1 , π 2 : G × G → G are the obvious projections and m : G × G → G is the group multiplication. This product • G agrees with the Adem-Ruan-Zhang product • ARZ on the orbifold K-theory of the inertia orbifold of [{pt}/G]. The ring (K G (G), • G ), or its complexification, is the fusion ring for the three-dimensional topological quantum field theory associated to the finite gauge group G, see [18] and [19]. Explicitly, given two elements in K G (G) = ⊕ (g) R(Z G (g)), the Pontryagin product is given by for any k ∈ G. The delocalized Chern character is a linear isomorphism of vector spaces. For an abelian group G,G = G × G. Then the Pontryagin product • G induces the convolution product on C G (G) in the first variable (3) The third one is the stringy product • on the orbifold K-theory of the orbifold [G/G] itself defined in Definition 4.3. Given two elements in K * orb ([G/G], C) ∼ = ⊕ (g) R(Z G (g)) ⊗ C, the stringy product on [G/G] is given by where V (g) • Z G (g) W (g) is the stringy product on K * orb ([{pt}/Z G (g)], C) discussed in Example 4.7.
The delocalized Chern character is a ring isomorphism, where the product * 2 on C G (G) is given by For an abelian group G, * 2 is the convolution product on C G (G) in the second variable A simple example like G = Z 2 shows that these three products are indeed different.  (Weighted projective spaces) Consider the weighted projective space W P(p, q), where p and q are coprime integers, which can be presented as the quotient of the unit sphere S 3 ⊂ C 2 by the S 1 -action e iθ (z 1 , z 2 ) = (e ipθ z 1 , e iqθ z 2 ).
As an orbisphere, W P(p, q) can be covered by two orbifold charts at singular points x = [1, 0] and y = [0, 1] with isotropy groups Z p and Z q respectively. Here Z w i is the cyclic subgroup of S 1 generated by the primitive w i -th root of unity.
The orbifold K-theory of W P(p, q) over C was given in [3] as K * orb W P(p, q), C ∼ = C[u]/ (1 − u) 2 ⊕ C(ζ p ) ⊕ C(ζ q ) where C(ζ p ) and C(ζ q ) are the p-th and q-th cyclotomic fields over C. We can apply Theorem 4.5, Remark 4.6 and Example 4.28 in [2] to find the stringy product on K * orb W P(p, q), C . Let α p ∈ C ⊕ C(ζ p ) β q ∈ C ⊕ C(ζ q ) be elements such that the delocalized Chern characters correspond to the constant function 1 on the twisted sectors corresponding to generators of Z p and Z q respectively. Then we have α p p = β q q = 1 − u, α p+1 p = β q+1 q = 0.

TWISTED CASES
As explained in Section 4.1, the Adem-Ruan-Zhang stringy product is defined for twisted K-theory of the inertia orbifold of a compact, almost complex orbifold. In this section, we explain how the constructions in Section 4 can be carried over to torsion twisted cases.
A twisting σ over an orbifold X is a principal P U (H)-bundle over X. Then there exists an orbifold atlas such that σ is represented by an U (1)-central extension of the canonically associated groupoid G[U ] = (G 1 ⇉ G 0 ). This central extension 1) called a U (1)-gerbe over (G 1 ⇉ G 0 ), is obtained from the P U (H)-valued cocycle on G[U ] and the U (1)-central extension of P U (H). Note that a gerbe connection is a connection θ on the U (1)-bundle R which is compatible with the groupoid multiplication on R. A curving for the connection θ is a 2-form B on G 0 such that the curvature of θ F θ = s * B − t * B ∈ Ω 2 (G 1 ).
Hence, the 3-form dB defines a closed 3-form Ω on the orbifold X, called the gerbe curvature of the gerbe R with a gerbe connection θ and a curving B. A twisting σ is called torsion if its associated gerbe is flat, that is, it has a gerbe connection and a curving whose gerbe curvature vanishes. By a standard construction (Proposition 3.6 in [43]), taking a refined orbifold atlas if necessary, the gerbe (5.1) always admits a gerbe connection and a curving.
The central extension (5.1) defines a complex line bundle over G 1 . Restricted to the inertia groupoid G, we get a complex line bundle over the inertia groupoid, denoted by L σ = ⊔ (g)∈T 1 L (g) −→ ⊔ (g)∈T 1 X (g) .

(5.3)
When the gerbe is equipped with a gerbe connection and curving, the property (5.2) implies that the induced connection on the complex line bundle L σ is flat. Hence, L σ = {L (g) } (g)∈T 1 is an inner local system defined in [34] and [39]. Recall that an inner local system over an orbifold is a flat complex orbifold line bundle L σ = ⊔ (g)∈T 1 L (g) −→ ⊔ (g)∈T 1 X (g) satisfying the following compatibility conditions (1) L (1) is a trivial orbifold line bundle with a fixed trivialization.
Let σ be a torsion twisting on an almost complex compact orbifold X. We assume that the twisted orbifold K-theory defined by the Grothendieck group of R-twisted vector bundles over X agrees with the twisted K-theory defined in Section 4.1. Letσ be the gerbe associated to σ with a gerbe connection and curving. Note that if the gerbe curvature ofσ is zero, then the twisted Chern character on the twisted orbifold K-theory K * orb (X, σ) constructed in [43] chσ : K * orb (X, σ) ⊗ C −→ H * (X, L σ ) = (g)∈T 1 H * (X (g) , L (g) ) (5.5) is an isomorphism of vector spaces over the complex coefficients. Geometrically, this twisted Chern character is constructed as follows. Let E be an R-twisted vector bundle over G 0 . Then the pull-back vector bundle e * E overG 0 admits a bundle isomorphism Φ defined by an element in End(e * E) ⊗ L σ . With respect to a locally constant trivialization, the Chern character ch Φ as in Section 4.2 can be extended to the twisted case to define a homomorphism ch Φ,σ : K * orb (X, e * σ) −→ H * (X, L σ ) such that the following diagram commutes K * orb (X, e * σ) ⊗ C ch Φ,σ / / H * (X, L σ ).
If e * σ is transgressive, then we can apply the Adem-Ruan-Zhang product on K * orb (X, e * σ) to define a stringy product on K * orb (X, σ) ⊗ C. Modifying the twisted Chern character in (5.5) as in Definition 4.4 and Theorem 4.5, we expect that K * orb (X, σ) ⊗ C with this string product is isomorphic to the twisted Chen-Ruan cohomogy H * (X, L σ ) using the Mayer-Vietoris exact sequence. We leave the details of the proof of this isomorphism to interested readers.